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Bethe-Salpeter equation for classical gravitational bound states

Tim AdamoRiccardo Gonzo
Dec 2022
摘要
The Bethe-Salpeter equation is a non-perturbative, relativistic and covariantdescription of two-body bound states. We derive the classical Bethe-Salpeterequation for two massive point particles (with or without spin) in a boundgravitational system. This is a recursion relation which involvestwo-massive-particle-irreducible diagrams in the space of classical amplitudes,defined by quotienting out by symmetrization over internal graviton exchanges.In this context, we observe that the leading eikonal approximation to two-bodyscattering arises directly from unitarity techniques with a coherent state ofvirtual gravitons. More generally, we solve the classical Bethe-Salpeterequation analytically at all orders by exponentiating the classical kernel inimpact parameter space. We clarify the connection between this classical kerneland the Hamilton-Jacobi action, making manifest the analytic continuationbetween classical bound and scattering observables. Using explicit analyticresummations of classical (spinless and spinning) amplitudes in momentum space,we further explore the relation between poles with bound state energies andresidues with bound state wavefunctions. Finally, we discuss a relativisticanalogue of Sommerfeld enhancement which occurs for bound state cross sections.
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